George Johnson's aritcle on "What is Math?" (2/10/98) contains lots of
interesting information about the "quasi-empirical" manner in which
mathematics is developed. The work of Lakoff and Nunez on what might be
called the underlying biology of human mathematical activity is particularly
striking. However, the inference from this work that mathematical truth once
arrived at has no status beyond that of being a human creation is quite
dubious. As an example, consider the theorem of elementary number theory
that every positive whole number can be obtained by adding together no more
than four perfect squares (e.g., 15=9+4+1+1) proved long ago by the French
mathematician Lagrange. Given the proof, who could doubt that this assertion
retains its validity for numbers far larger than any directly accessible to
human or computer manipulation and indeed will still be true long after the
human race itself has vanished? This has nothing to do with the hoary
non-question of whether numbers somehow "exist" in a Platonic world of ideal
concepts, and even less to do with the manner (made much of in Johnson's
article) in which we write numbers using 10 digits.
Martin Davis
[Note: "Nunez" has accents, correctly shown in the article.]
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Martin Davis is Professor Emeritus at New York University and is currently a
Visiting Scholar at the University of California in Berkeley.